Problem: s involving definite integrals (algebraic) AP.CALC: CHA‑4 (EU), CHA‑4.D (LO), CHA‑4.D.1 (EK), CHA‑4.D.2 (EK), CHA‑4.E (LO), CHA‑4.E.1 (EK) Google Classroom Facebook Twitter Email You might need: Calculator Problem A joint is moving at an angular velocity of $0.2e^{0.2t}$ radians per second (where $t$ is the time in seconds since the joint was at rest). Through how many radians does the joint move between $t=1$ and $t=3$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $e^{0.4}$ (Choice B) B $e^{0.6}-e^{0.2}$ (Choice C) C $0.2(e^{0.6}-e^{0.2})$ (Choice D) D $0.2e^{0.4}$
Answer: Letting $\theta(t)$ be the angular displacement after $t$ seconds, we are given that $\theta'(t)=0.2e^{0.2t}$. We want to find $\theta(3)-\theta(1)$. According to the Fundamental Theorem of Calculus, $\begin{aligned} \theta(3)-\theta(1)&=\int_{1}^{3} \theta'(t)\,dt \\\\ &=\int_{1}^{3}(0.2e^{0.2t})\,dt \end{aligned}$ $\int_{1}^{3}(0.2e^{0.2t})\,dt = e^{0.6}-e^{0.2}$ In conclusion, between $t=1$ and $t=3$ the joint moves through $e^{0.6}-e^{0.2}$ radians.